MNRAS accepted, 1–17 (2017) Preprint 24th September 2018 Compiled using MNRAS LATEX style file v3.0

Real- and redshift-space halo clustering in f (R) cosmologies

Pablo Arnalte-Mur, 1,3,4? Wojciech A. Hellwing 2,3,5 and Peder Norberg 3,61Observatori Astronòmic de la Universitat de València, C/Catedràtic José Beltrán, 2, 46980 Paterna, Spain2Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK3Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK4Departament d’Astronomia i Astrofísica, Universitat de València, 46100 Burjassot, Spain5Janusz Gil Institute of Astronomy, University of Zielona Góra, ul. Szafrana 2, 65-516 Zielona Góra, Poland6Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK

Accepted for publication in MNRAS - 19th January 2017

ABSTRACTWe present two-point correlation function statistics of the mass and the halos in the chameleonf (R) modified gravity scenario using a series of large volume N-body simulations. Three dis-tinct variations of f (R) are considered (F4, F5 and F6) and compared to a fiducial ΛCDMmodel in the redshift range z ∈ [0, 1]. We find that the matter clustering is indistinguish-able for all models except for F4, which shows a significantly steeper slope. The ratio ofthe redshift- to real-space correlation function at scales > 20 h−1 Mpc agrees with the linearGeneral Relativity (GR) Kaiser formula for the viable f (R) models considered. We considerthree halo populations characterized by spatial abundances comparable to that of luminousred galaxies (LRGs) and galaxy clusters. The redshift-space halo correlation functions of F4and F5 deviate significantly from ΛCDM at intermediate and high redshift, as the f (R) halobias is smaller or equal to that of the ΛCDM case. Finally we introduce a new model inde-pendent clustering statistic to distinguish f (R) from GR: the relative halo clustering ratio – R.The sampling required to adequately reduce the scatter in R will be available with the adventof the next generation galaxy redshift surveys. This will foster a prospective avenue to obtainlargely model-independent cosmological constraints on this class of modified gravity models.

Key words: gravitation – methods: data analysis – cosmology: theory – dark matter – large-scale structure of Universe

1 INTRODUCTION

The hot relativistic big-bang Λ-cold dark matter (ΛCDM) cosmo-logy is a very successful standard model of cosmology. It passesa tremendous amount of observational tests, from properties of theCMB (e.g. Hinshaw et al. 2013), large-scale clustering of galaxies(e.g. Cole et al. 2005; Eisenstein et al. 2005; Zehavi et al. 2011;Alam et al. 2016), weak and strong lensing (e.g. Bartelmann &Schneider 2001; Schrabback et al. 2010; Suyu et al. 2013) to prop-erties of galaxy clusters, galaxies and their satellites in the nearbyUniverse (e.g. Mandelbaum et al. 2006; Allen et al. 2011; Wojtaket al. 2011; Guo et al. 2015; Umetsu et al. 2016). The minimumset of parameters describing this simple scenario has been nowestablished to a remarkable precision (Planck Collaboration et al.2016). Despite its undeniable success, the standard ΛCDM modelsuffers from serious theoretical problems. The model explains theobserved late-time acceleration of the Universe (Riess et al. 1998;Perlmutter et al. 1999) by attributing it to a very low positive valueof Einstein’s cosmological constant, Λ. One of the main shortcom-ings of this approach comprise the fact that the only known possible

? E-mail: pablo.arnalte@uv.es

physical explanation of the non-zero Λ is the zero-point energy ofvacuum quantum fluctuations. However, quantum theory predictsa natural value for Λ that is many orders of magnitude larger thanthe actual value that is compatible with observations (for an excel-lent discussion see, e.g. Carroll 2001, and references therein). Theunavoidable conclusion is that one of the fundamental ingredientsof the equations describing the evolution of the cosmological back-ground is lacking a clear physical interpretation. In addition Gen-eral Relativity (GR), as any working physical theory, itself needs tobe continuously tested on all scales and regimes accessible throughexperiments and observations (Will 2014).

The conceptual problems of ΛCDM have motivated a numberof theoretical modifications to the standard model, which can pro-duce the observed late-time acceleration of the Universe by meansof different physical mechanisms. The rich literature on the sub-ject can be divided broadly into two distinct categories. In the first,it is postulated, that the acceleration is produced by a dynamic-ally evolving background scalar field (for a solid review of thesubject see Copeland et al. 2006). These models, usually invok-ing a minimally coupled scalar field, are collectively dubbed asdark energy. The second category consists of theories where theaccelerated expansion is a manifestation of the modifications to the

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Einstein-Hilbert action integral. I.e. they implement modificationsto the theory of GR, an otherwise fundamental building block ofmodern cosmology (Brax et al. 2008; Clifton et al. 2012; Koy-ama 2016). The latter class of models are the so-called modifiedgravity (MOG) models. Here the late-time acceleration is fueled byextra terms appearing in the cosmic Lagrangian and which act asan ‘effective Lambda’ term. Thus, in this approach a mechanismthat would set the usual cosmological constant to exactly zero isneeded. Since such a mechanism has not yet been discovered, thisapproach should not be regarded as an attempt to construct a newfundamental theory of gravity, but rather an effort to probe the richphenomenology of infrared modifications to GR, with non-trivialeffects on cosmological scales. MOG models, in principle, can beconstructed in many different ways. In the recent years, one of thebroadly investigated models, that fall into the MOG category, is theso-called f (R) gravity theory. In this case the accelerated expan-sion is produced by an extra term replacing Λ in the action integral.This term consists of a non-linear function f taking as an argumentthe curvature scalar R (Navarro & Van Acoleyen 2007; de Felice &Tsujikawa 2010; Sotiriou & Faraoni 2010). This class of models ex-hibit rich and interesting new physics. In addition to producing late-time acceleration they admit for a non-negligible fifth force actingon small and intermediate cosmological scales (i.e. much smallerthan the horizon, cH−1

0 ). This non-trivial and intrinsically non-linear fifth force can manifest itself in deviations of the large andsmall-scale clustering of galaxies and matter from the standard GRpicture. In other words: in f (R) gravity one can have an universeexhibiting GR, or ΛCDM-like, expansion history but admitting, atthe same time, a different history of growth of structures (Faulkneret al. 2007; Brax et al. 2008; Oyaizu et al. 2008; Li et al. 2013).

Any potentially successful MOG theory is required to not onlypredict a global expansion history compatible with observations,but also needs to pass stringent local tests of gravity. The lattercome from observed orbital dynamics in the Solar System (e.g.Chiba et al. 2007; Hu & Sawicki 2007; Berry & Gair 2011), pulsartiming (Brax et al. 2014) and as of recently the physics of grav-itational waves emitted during black hole mergers (Raveri et al.2015; Abbott et al. 2016). See also Berti et al. (2015) for a dis-cussion of other astrophysical test of modified gravity. In the mostgeneral class of f (R) theories the fifth-force can freely propagatewhenever there is a gradient of the f (R) scalar field (also calledthe scalaron). Thus, if this model wants to stay compatible withthe local gravity tests, it needs to implement a mechanism for sup-pressing the fifth force in high-density regions, like our Solar Sys-tem or neutron star binaries. In f (R) theories this is accomplishedby a convenient choice of the f (R) function that give rise to the so-called chameleon mechanism (Khoury & Weltman 2004; Brax et al.2008). The chameleon mechanism makes the scalaron very massivein spatial regions of high local curvature (density), this leads to aneffective suppression of any fifth-force propagation. Contrastingly,in regions with a low local density, the field is light and admitsthe propagation of the scalar fifth-force. The effectiveness of thechameleon suppression is moderated by the local density field. Thismakes this mechanism to be intrinsically environment dependentand thus highly non-linear in its nature. Consequently, in this scen-ario, one can have regions of low cosmic density (such as e.g. cos-mic voids) in which the fifth-force strongly affects the dynamicsand clustering of galaxies, as well as regions with higher density,where the theory can effectively behave as the classical GR. Asthe degree of non-linearity in both matter and scalar cosmic fieldsincreases fast during cosmic evolution, it quickly renders predic-tions of simple linear and weakly non-linear perturbation theory

unreliable (e.g. Hellwing 2015). Because of this, the use of N-bodycomputer simulations is essential for forecasting reliable and ac-curate predictions. However the same very non-linear nature makessuch simulations much more challenging and more expensive thanstandard GR simulations. In the recent years there has been a sig-nificant progress in the development of modified gravity N-bodysolvers (e.g. Oyaizu 2008; Schmidt et al. 2009b; Zhao et al. 2011;Li et al. 2012a; Puchwein et al. 2013; Llinares et al. 2008; Llinares& Mota 2013, 2014; Winther et al. 2015; Bose et al. 2016). As anoutcome, modern codes are not only capable of running big volumeand high-resolution simulations, but also have attained the accuracyneeded for the precision cosmology era of the current and forth-coming galaxy surveys, such as Euclid (Laureijs et al. 2011), theDark Energy Spectroscopic Instrument survey (DESI, Levi et al.2013), or the Javalambre-Physics of the Accelerated Universe As-trophysical Survey (J-PAS, Benitez et al. 2014). Thanks to this, it isnow possible to study the galaxy, halo and matter clustering prop-erties of f (R) gravity models with sufficient resolution.

In general, we can expect that in f (R) models the modific-ations to GR will manifest themselves as a modified history ofgrowth of structures, and thus will also affect the galaxy cluster-ing and dynamics. It has been shown in the literature that indeedthis class of models exhibit higher amplitude of matter power spec-trum at small and intermediate scales (i.e. . 20 h−1 Mpc) (Oyaizuet al. 2008; Gil-Marín et al. 2011; Li et al. 2013), and even on largerscales for the case of higher-order clustering amplitudes (Hellwinget al. 2013). Dark matter clustering in redshift space is also char-acterized by stronger Finger-of-God (FOG) effects at small scales(Jackson 1972), which is accompanied by more pronounced Kaisereffect (Kaiser 1987; Hamilton 1992) at larger scales (Jennings et al.2012). The stronger FOG, which leads to more effective small-scale power damping in redshift space, is a manifestation of dy-namics enhanced by the fifth force. This effective enhancementwas also shown to be predicted, as a prominent MOG ’smokinggun’ feature, for the galaxy/DM halo velocity field (Hellwing et al.2014). Other studies have shown that f (R) models can lead to dif-ferent predictions for density profiles and size of cosmic voids (Liet al. 2012b; Cai et al. 2015), modified stellar evolution (Sakstein2015), or several characteristics of galaxy clusters: number counts(Schmidt et al. 2009b), X-ray or lensing radial profiles (Wilcoxet al. 2016) and measured gas fractions (Li et al. 2016).

All the above mentioned effects of MOG in general shouldmanifest themselves in observations as deviations from the GR-based predictions. However the highly non-linear character of thegalaxy formation process makes it very difficult to foster obser-vational predictions with respect to GR/MOG differences. Highlyenergetic processes, such as star formation feedback and ActiveGalactic Nuclei (AGN) feedback affect the matter distribution up toscales of 20 h−1 Mpc (e.g. van Daalen et al. 2011, 2014; Hellwinget al. 2016). It was shown that, when matter clustering is concerned,the baryonic feedback effects are degenerate with enhanced cluster-ing predicted by pure collisionless simulations of f (R) (Puchweinet al. 2013). Therefore a good strategy aimed to find a clean f (R)signature is to look at both larger-scales and at more massive ha-loes. Here one can expect that the baryonic effects should be rel-atively weaker, giving hope of reducing the baryonic-MOG effectsdegeneracy.

These previous works have studied the expected changes inthe growth of structures in f (R) models by analysing the changesin different clustering properties of the DM density field. However,in order to be able to compare the models with observational datafrom galaxy surveys, one needs to obtain a prediction for the clus-

MNRAS accepted, 1–17 (2017)

Halo clustering in f (R) cosmologies 3

tering of galaxies. This involves studying possible differences inthe biasing mechanism between ΛCDM and the f (R) models. Inprinciple this would require the modelling of the galaxy forma-tion process in the f (R) theory. A first step in this direction is tostudy the clustering properties of DM haloes, as the bias of galax-ies is closely related to the bias of the haloes in which they reside.Moreover, when we restrict the study to the most massive haloesand linear or quasi-linear scales, the clustering of DM haloes is agood proxy for the clustering of the corresponding central galax-ies. A complementary approach was followed by He et al. (2016)who used the sub-halo abundance matching technique to study theclustering of galaxies in the f (R) model at small non-linear scales(r 6 6 h−1 Mpc).

The aim of this work is therefore to characterize the clusteringproperties of DM haloes in a set of f (R) models and compare themto the ΛCDM model. We explain how the clustering of massive ha-loes is affected by f (R) enhanced dynamics in both real and redshiftspace. We also conduct our study for a range of cosmic times, aim-ing to find the epoch of cosmic evolution at which the relative dif-ferences between the models are strongest. Our ultimate goal is toconfront the theoretical predictions with observations from galaxyredshift surveys. Hence, when selecting our samples and definingour clustering observables, we try to match what could be feasiblewhen using real data. Following this approach, we define a newstatistic that can be easily measured from observations, and whichcan potentially help discriminate between GR and f (R) cosmolo-gies in the real Universe.

This paper is organized as follows. In section 2 we give a briefdescription of both the physical set-up of the f (R) model and ofthe numerical simulations used in this work. The clustering statist-ics and the definition of the different halo samples that we use aredescribed in section 3. Section 4 concerns the results of our ana-lysis, while in section 5 we discuss potential observational cluster-ing tests using the new clustering ratio statistic. Finally in section 6we give our conclusions.

2 THE f (R) GRAVITY THEORY AND SIMULATIONS

Here we briefly introduce the physical set-up and basic propertiesof the f (R) modified gravity model accompanied by a descriptionof the numerical structure formation simulations used in this work.

2.1 The f (R) gravity theory

The f (R) gravity (Carroll et al. 2005) is an extension of GR thathas been extensively studied in the literature in the past few years.The main properties of the model are widely known, hence we willfocus here on only a very brief introduction of this theory, refer-ring the reader for more details to the rich literature on the subject(see e.g. Sotiriou & Faraoni 2010; de Felice & Tsujikawa 2010, fordetailed reviews).

The theory is obtained by substituting the Ricci scalar R in theEinstein-Hilbert action with an algebraic function f (R),

S =

∫d4 x√−g

M2

Pl

2[R + f (R)

]+Lm

. (1)

Here MPl is the reduced Planck mass, M−2Pl = 8πG, G is Newton’s

constant, g the determinant of the metric gµν andLm the Lagrangiandensity for matter and radiation fields (including photons, neutri-nos, baryons and cold dark matter). By designing the functionalform of f (R) one can fully specify a f (R) gravity model.

Varying the action, eq. (1), with respect to the metric field gµν,one obtains the modified Einstein equation

Gµν + fRRµν − gµν

[12

f (R) − fR

]− ∇µ∇ν fR = 8πGT m

µν, (2)

where Gµν ≡ Rµν −12 gµνR is the Einstein tensor, fR ≡ d f /dR, ∇µ is

the covariant derivative compatible with gµν, ≡ ∇α∇α and T mµν is

the energy momentum tensor of matter and radiation fields. Eq. (2)is a fourth-order differential equation, but can also be consideredas the standard second-order equation of GR with a new dynam-ical degree of freedom, fR, the equation of motion of which can beobtained by taking the trace of eq. (2)

fR =13

(R − fRR + 2 f (R) + 8πGρm) , (3)

where ρm is the matter density. This new degree of freedom fR isthe scalaron mentioned earlier.

Our analysis here is mainly concerned with large-scale struc-tures, which are much smaller than the Hubble scale. Since the timevariation of fR is very small in the models to be considered below,we shall work in the quasi-static limit by neglecting the time deriv-atives of fR. It has been shown that by adopting this approximation,the resulting modelled dynamics of the scalar and matter fields de-viates negligibly from the true dynamics (Bose et al. 2015). Underthis limit, the fR equation of motion, eq. (3), reduces to

~∇2 fR = −13

a2[R − R + 8πG (ρm − ρm)

], (4)

where ~∇ is the three dimensional gradient operator, and the overbartakes the background ensemble average of a quantity.

Similarly, the Poisson equation, which governs the behaviourof the gravitational potential Φ, simplifies to

~∇2Φ =16πG

3a2 (ρm − ρm) +

16

a2[R − R

], (5)

by neglecting terms involving time derivatives of Φ and fR, andusing eq. (4) to eliminate ~∇2 fR.

The above considerations foster two ways in which the scal-aron field can affect cosmology: (i) the background expansion ofthe Universe can be modified by the new terms in eq. (2) and (ii)the relationship between the gravitational potential Φ and the mat-ter density field is modified, which can affect the matter cluster-ing and growth of density perturbations. Clearly, when | fR| 1,we have R ≈ −8πGρm (see eq. (4)) and thus eq. (5) reduces tothe usual Poisson equation; when | fR| is large, we will have rather|R − R| 8πG|ρm − ρm| and then eq. (5) simplifies to the stand-ard Poisson equation, but with G rescaled by 4/3. The value 1/3is the maximum intensification factor of gravity in f (R) models,independent of the specific functional form of f (R). The choice off (R), however, is crucial because it determines the scalaron dynam-ics and therefore when and on what scales the enhancement factorchanges from 1 to 4/3. Scales much larger than the range of themodification to Newtonian gravity mediated by the scalaron field(i.e., the Compton wavelength of fR) are unaffected and gravity isnot enhanced there, while on small scales, depending on the envir-onmental matter density, the 1/3 enhancement may be fully real-ized. This results in a scale-dependent modification of gravity andtherefore a scale-dependent growth rate of structures already at thelinear theory level (Koyama et al. 2009).

2.1.1 The chameleon mechanism

The gravity and Newtonian dynamics passes stringent tests comingfrom the Solar System observations, and so any 4/3 force enhance-

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4 Arnalte-Mur, Hellwing & Norberg

ment factor related to f (R) needs to avoid high-density regions asour Solar System. The theory achieve this by implementing the so-called chameleon screening mechanism.

The basic idea of the chameleon mechanism is the follow-ing: the modifications to Newtonian gravity can be considered asa fifth force mediated by the scalaron field fR. Because the scalaronis massive, this extra force experiences a Yukawa-type potential.Hence the enhanced gravity is decaying exponentially as exp(−mr),in which m is the scalaron mass, as the distance r between two testmasses increases. In high matter density environments, m is veryheavy and the exponential decay causes a strong suppression of theforce over distance. In reality, this is equivalent to setting | fR| 1in high density regions because fR is the potential of the fifth force,and this leads to the GR limit as we have discussed above.

Consequently, the functional form of f (R) is crucial in de-termining whether the fifth force can be sufficiently suppressed inhigh density environments. In this work we consider the f (R) Lag-rangian proposed by Hu & Sawicki (2007), for which

f (R) = −M2c1

(−R/M2

)n

c2(−R/M2)n

+ 1, (6)

where M2 ≡ 8πGρm0/3 = H20ΩM, with H being the Hubble ex-

pansion rate and ΩM the present-day fractional density of matter.Throughout the paper a subscript 0 always denotes the present-day(a = 1, z = 0) value of a quantity. It was shown by Hu & Sawicki(2007) that | fR0| . 0.1 is already sufficient to pass the Solar systemconstraints, but the exact constraint depends on the behaviour of fR

in galaxies and pulsating stars as well (Sakstein 2013, 2015). At thebackground level the scalaron fR always sits close to the minimumof the effective potential, therefore for the smooth scalar field wehave (Brax et al. 2012):

−R ≈ 8πGρm − 2 ¯f (R) = 3M2(a−3 +

2c1

3c2

)(7)

The Hu-Sawicki model we consider is fixed by requesting that thebackground expansion history matches that of ΛCDM. Thus, weset

c1

c2= 6

ΩΛ

ΩM(8)

where ΩM and ΩΛ are respectively the present-day fractional en-ergy densities of the matter and dark energy. The simulation weuse in this work use WMAP3 cosmological background parameters(Spergel et al. 2007) (see Table 1). Using ΩΛ = 0.76 and ΩM = 0.24and eq. (7) gives |R| ≈ 41M2 M2 at late times. Using this approx-imation simplifies the expression of the scalaron to the followingform

fR ≈ −nc1

c22

(M2

−R

)n+1

. (9)

The above considerations show that once a ΛCDM background isfixed, our chosen f (R) model is completely specified by the twofree parameters: n and c1/c2

2. Henceforth, the ratio c1/c22 is also

fixed by the averaged background value of the scalaron, fR0, at z =

0. This yields

c1

c22

= −1n

[3(1 + 4

ΩΛ

ΩM

)]n+1

fR0. (10)

Thus the choice fR0 and n fully specifies our model.The particular f (R) set-up we consider here have very interest-

ing cosmological properties. At small scales in regions where thelocal density is high the enhanced gravity will be suppressed and

the dynamics will be Newtonian. Hence we can expect that orbitalsatellites and halo close interactions will be very similar as in GR.However in regions exhibiting low densities, such as e.g. cosmicvoids, the modified dynamics should affect both halo and galaxyclustering and velocities. We specifically consider three flavours ofthe Hu-Sawicki f (R) model with fixed n = 1, that differ in thepresent-day mean (background) scalaron value | fR0| = 10−4, 10−5

and 10−6. We dub the models F4, F5 and F6 consequently. Thesethree models cover the portion of the f (R) parameter space thatproduce interesting cosmological effects and is still compatiblewith extragalactic observations. While F5 and F6 are so far in abroad agreement with the cosmological observations, F4 howeveris already in a strong tension with observations of cluster num-ber counts (Schmidt et al. 2009b; Ferraro et al. 2011; Lombriseret al. 2012; Cataneo et al. 2015) or weak lensing (Harnois-Dérapset al. 2015; Liu et al. 2016). Thus we shall use F4 results just as anextreme example of effects induced by only weakly-screened fifthforce.

2.2 Cosmological f (R) simulations used in this work

In this work we use the f (R) simulations introduced in Li et al.(2013). Most of the previous work however focused on DM densityfields only. Here we are very much interested in clustering proper-ties of DM haloes (and ultimately galaxies). For that reason wehave applied Rockstar, a phase-space Friends-of-Friends (FOF)halo finder (Behroozi et al. 2013). We kept all the haloes that con-tained at least 100 DM particles, hence this sets our minimal halomass limit to Mmin = 2.09× 1013 h−1 M. Further on we recomputethe FOF halo mass using a proper virial mass definition. For thevirial mass we use M200, i.e. the mass contained in a sphere of ra-dius r200 centred on a halo, such that the average overdensity insidethe sphere is 200 times the critical closure density, ρc ≡ 3H2/8πG.Our adopted mass-cut left us with ∼ 106 haloes at z = 0 for eachinitial condition realization. Thus the upper-limit on our spatialnumber density of objects is n = 3 × 10−4 h3 Mpc−3 at z = 0 andcorrespondingly smaller at higher redshifts. Our simulations usea computational domain of 1500 h−1 Mpc size. Following the ana-lyses by other authors of the importance of both finite-volume ef-fects (e.g. Colombi et al. 1994) and sparse-sampling (e.g. Szapudi& Colombi 1996) we adopt conservative limits on the minimal andmaximal scales that we trust. For a minimum scale we adopt a limitof 3 × 2πk−1

Nyq ' 10 h−1 Mpc, where the Nyquist frequency for thesimulations is kNyq = 2.14 h Mpc−1. We take as the maximum scaleto study 1/10 × Lbox ' 150 h−1 Mpc, as we expect larger scales tobe affected by the finite volume effects. Finally we will focus ouranalysis on 4 snapshots taken consecutively at z = 0, 0.25, 0.66 and1.0. Previous studies (Hellwing et al. 2013) have shown that forthose times the differences between GR and f (R) clustering are ex-pected to be the largest. We list other details of the simulations usedhere in Table 1.

3 ANALYSIS OF HALO CLUSTERING IN N-BODYSIMULATIONS

The astronomical observations that provide the data characteriz-ing the clustering of matter at large scales contain information onlyabout the luminous stellar matter distribution in our Universe. Con-temporary galaxy redshift catalogues contain positions of millionsof galaxies, observed over large parts of the sky and over vast dis-tances (redshifts). Ideally one would like then to study the clus-

MNRAS accepted, 1–17 (2017)

Halo clustering in f (R) cosmologies 5

Table 1. Main properties of the simulations used in this work, for moredetails please see Li et al. (2013); Hellwing et al. (2013).

Models ΛCDM, F6, F5, F4Number of realizations 6Box size Lbox = 1500 h−1 MpcNumber of particles Np = 10243

Particle mass mp ' 2 × 1011 h−1 MNyquist frequency kNyq = 2.14 h Mpc−1

Force resolution ε = 22.9 h−1 kpc

Cosmological parameters:total matter density ΩM = 0.24dark energy density ΩΛ = 0.76baryonic matter density Ωb = 0.04181dimensionless Hubble parameter h = 0.73tilt factor of the initial power spectrum ns = 0.958power spectrum normalization σ8 = 0.77BAO peak scale (linear theory) rBAO ' 113 h−1 Mpc

tering of galaxies in various competing cosmological models. Thisrequires however introduction of another component into a theoryunder investigation, namely the galaxy formation model.

Various techniques exist that allow for galaxy formation mod-elling, the semi-analytic models (SAMs) (for a review see Baugh2006), hydrodynamical simulations (Vogelsberger et al. 2014;Schaye et al. 2015) and abundance matching (Kravtsov et al. 2004;Moster et al. 2010), to just name a few. However all the existingtechniques were developed and tested self-consistently only for theΛCDM model. Application and extrapolation of such modellingto MOG models is neither straightforward nor simple (Fontanotet al. 2013). In addition the existing ΛCDM galaxy formation mod-els are still subject of intensive scrutiny (Contreras et al. 2013), asour understanding of the importance and interconnection of all thecomplicated baryonic feedback processes is far from being full andcomplete (see e.g. Schaye et al. 2010; McCarthy et al. 2010; Fabjanet al. 2010; McCarthy et al. 2011; Puchwein & Springel 2013). Inaddition, the strength and the environmental dependence of the ad-ditional fifth-force of the f (R) model impact the galaxy clusteringin a way that is degenerated with strong baryonic feedback invokedby AGNs and galaxy winds (Puchwein et al. 2013).

Taking into account all the difficulties mentioned above andthe challenges connected with galaxy formation, we decide to fol-low a simpler approach. We use DM haloes and their clusteringproperties as proxies for galaxy clustering. Haloes are well definedobjects (both in ΛCDM and f (R)), and as such can be straightfor-wardly identified and extracted from N-body simulations (Knebeet al. 2011).

We expect that in f (R) gravity the galaxy formation mechan-ism and processes involved can, in principle, take largely differentcharacter than in ΛCDM. However, if we restrict the analysis to asample of very luminous galaxies, the situation is simpler. In thiscase, the fraction of satellite galaxies is very small (e.g. Zheng et al.2009), so we can assume that most of the galaxies are located at thecentres of massive DM haloes and the galaxy clustering propertieswill follow closely those of the host haloes. This is certainly a validapproximation if we constrain ourselves to sufficiently large scales(i.e. the two-halo term limit, > 10 h−1 Mpc). Moreover, for this typeof galaxy samples, it is possible to remove the effect of the satel-lite galaxies from clustering measurements (Reid, Spergel & Bode2009). Going further to the high-mass end of the mass function,DM haloes correspond to galaxy groups or clusters, which can be

identified from galaxy surveys (see, e.g., Koester et al. 2007; Ro-botham et al. 2011; Ascaso et al. 2015). In this regime, the cluster-ing of DM haloes in different models can be directly compared tothat of observed groups.

To characterize the clustering of matter and haloes (in positionand redshift space) at different scales and epochs we use a basic2-point statistic: the two-point correlation function, ξ(r). This isdefined as (Peebles 1980) the excess probability (with respect toa Poisson process) of finding two haloes contained in two volumeelements dV1 and dV2 at a distance r:

dP12(r) ≡ n2[1 + ξ(r)]dV1dV2 , (11)

where n is the mean halo (galaxy) number density.In general, the halo 2-point correlation function will depend

on the selected halo population (H), the redshift (z), and the cosmo-logical model (M) considered, which we denote as ξ(r|z,H ,M).Because a density perturbation in an expanding universe needs topass a certain threshold value δc

1 in order to be able to collapseand form a gravitationally bound structure (i.e. a halo), the haloesare biased tracers of the underlying smooth matter density field (seee.g. Fry & Gaztanaga 1993). We parametrize this through a simplelinear relation:

ξ(r|z,H ,M) = b2(r|z,H ,M)ξm(r|z,M) , (12)

where b(r|z,H ,M) is the linear bias parameter and ξm(r) is thecorrelation function of the matter density field.

Generally, we can expect that the main differences betweenΛCDM and f (R) halo clustering, will arise due to: (i) differentamplitudes of the matter correlation function, ξm, at the same scaler, and (ii) deviation in the bias parameter, which will be driven byboth the departure in the halo mass - bias relation and by the differ-ences in the selection of a particular halo population. We will studythe clustering of haloes in redshift space, as this corresponds towhat would be available from observations. Therefore, further dif-ferences can originate from changes in the effects of redshift-spacedistortions in different gravity models.

We present in section 3.1 the method we use to measure thehalo correlation function in the simulations. In section 3.2 we showthe halo mass functions obtained in the simulations and we alsoexplain the approach used to select the different halo populationswe analyse in section 4.

3.1 Estimation of the correlation function in the simulations

In this work, we estimate the correlation function for differenttracers (haloes or DM particles) extracted from N-body simula-tions. This means that the selection function in all cases is com-plete, isotropic and homogeneous. Moreover, as the volume is abox with periodic boundary conditions, we do not need to correctfor any edge effects. Therefore, we obtain the correlation functionin each case using the simple estimator:

ξ(r) =DD(r)Nnv(r)

− 1 , (13)

1 The critical density threshold for collapse takes different values in variouscosmologies. For ΛCDM δc ' 1.673 (Peebles 1980; Weinberg & Kami-onkowski 2003). For f (R) this number is no longer universal as the fifth-force has an environmental and scale dependence (Li & Efstathiou 2012).Schmidt et al. (2009a) have shown for example that, when the chameleoneffect is ignored, the value for F4 is δc ' 1.692.

MNRAS accepted, 1–17 (2017)

6 Arnalte-Mur, Hellwing & Norberg

where DD(r) is the number of pairs of tracers with separation in therange [r, r + ∆r], N is the total number of tracers in the sample, n istheir number density, and v(r) is the volume of a spherical shell ofradius r and width ∆r,

v(r) =4π3

[(r + ∆r)3

− r3]. (14)

We use in all cases bins in separation of width ∆r = 8 h−1 Mpc. Thissimple estimator is much faster than the estimators usually used forreal data, such as that from Landy & Szalay (1993), as in this casewe do not need to use an auxiliary random sample to correct forthe selection function or edge effects. We checked that we obtainedidentical results when using the Landy & Szalay (1993) estimatorfor our calculations.

We compute, for each tracer, the correlation function separ-ately in each of our six realizations and take as our value for thecorrelation function of this tracer the mean of these six estimations.To estimate the corresponding error we use the standard error on themean over the ensemble of six realizations. This is a conservativeerror estimation (Szapudi & Colombi 1996) that takes into accountthe contributions of both the cosmic variance and the shot noise.Although cosmic variance is the main source of uncertainty for thedark matter correlation function, shot noise is also important whenwe consider samples of massive haloes, with low number density.As we are combining here our six realizations, the statistical errorwe obtain would correspond to that achievable by an ideal surveycovering a volume of V = 6 ×

(1500 h−1 Mpc

)3' 20 h−3 Gpc3.

When we compute the dark matter correlation function (in sec-tion 4.1) we use a random subsample containing ' Np/1000 DMparticles in each realization. This subsample is obtained by ran-domly selecting particles from the ID list, so that all the populationproperties are sampled uniformly. This avoids the need for a prohib-itive computation time, while not affecting the results, as the errorsin the resulting sample are still dominated by cosmic variance, andnot by shot noise. For comparison, the resulting number density ofDM particles used in our calculations is still ∼ 10 times larger thanthat of the densest halo sample used (see below).

3.2 Halo mass function and selection of halo populations

Before discussing the halo populations selected for our analysis,we need to consider the halo mass function of our simulations.This is shown for four different times (z = 0, 0.25, 0.66, 1.0) in thefour panels of Fig. 1. It is quite obvious that our simulations suf-fer significantly from numerical shot-noise effects at the low massend. Due to limited mass and spatial resolution of the simulations,the small-mass haloes suffer from the well known overmerging ef-fect (Klypin et al. 1999a,b; Moore et al. 1999). Thus the numberdensity of small-mass haloes is underestimated. This is clearly in-dicated by the change of slope of the halo mass functions aroundM200 ∼ 2× 1013 h−1 M. For GR, this mass roughly corresponds ton = 10−4 h3 Mpc−3 at z = 0, and to n = 3 × 10−5 h3 Mpc−3 at z = 1.Although it seems that the magnitude of the resolution effects isvery similar in all the models we study (Winther et al. 2015), for thesake of fair comparison we decide to restrict ourselves to this lim-iting number density as the highest one we consider. As the massfunction is always larger for the f (R) models than for GR, this lim-iting n should be sufficient for all our models.

The additional analysis of the plots in Fig. 1 reveals the beha-viour already found by other authors (Schmidt et al. 2009a; Li et al.2013; Hellwing et al. 2013). The largest deviation with respect tothe ΛCDM case is observed, as expected, for the F4 model. In this

Table 2. Properties of the halo samples used in this work. In each case,we list the minimum halo mass Mmin used to obtain the required numberdensity n for a given redshift z.

Halo n Mmin [1013 h−1 M]z population [ h3 Mpc−3] GR F6 F5 F4

0 H1 3 × 10−5 4.23 4.31 5.15 5.36H2 10−5 8.94 8.98 10.43 11.39H3 3 × 10−6 17.50 17.52 19.41 22.07

0.25 H1 3 × 10−5 3.77 3.81 4.54 4.71H2 10−5 7.58 7.62 8.77 9.63H3 3 × 10−6 14.51 14.51 16.02 18.26

0.66 H1 3 × 10−5 2.81 2.83 3.25 3.48H2 10−5 5.23 5.26 5.84 6.51H3 3 × 10−6 9.57 9.59 10.26 11.85

1.00 H1 3 × 10−5 1.86 1.91 2.20 2.60H2 10−5 3.73 3.75 4.00 4.65H3 3 × 10−6 6.53 6.55 6.80 8.10

case, the mass function already shows a significant deviation fromΛCDM at z = 1, with this deviation slightly increasing towards thelargest halo masses. The F5 model, with a more efficient screening,experiences a more complicated behaviour of the halo mass func-tion. Due to the screening, the deviation is very small at the high-mass end, and we observe that the mass at which the halo abund-ances depart from the fiducial model is growing with time. For boththe F4 and F5 models, for the range of halo masses not strongly af-fected by the screening, the relative departure of the halo numberdensity from the ΛCDM case tends to shrink with time. This re-flects the known effect that initially the ΛCDM model experiencesa structure formation that is retarded with respect to the MOG mod-els, but at late evolutionary stages the halo growth slows down inthe fifth-force cosmologies and so the ΛCDM is able to shrink theinitial gap (Hellwing et al. 2010). This is mostly due to the relativescarcity of small haloes available for mergers, that is handicappingthe halo mass growth via mergers at late times in f (R). Finally, forthe F6 model we do not observe any significant deviation from theΛCDM case.

We select different halo populations from our simulations bydefining a series of threshold samples, i.e. selecting haloes withmass above a certain value Mmin. Since for massive haloes the virialmass - luminosity relation (or mass-to-light ratio) is monotonic anddeterministic (Moster et al. 2010), such cuts are equivalent, on afirst approximation, to a sample of galaxies selected by luminosity.However, as the virial halo mass is not an observable, a selectionwith a fixed Mmin can not be directly replicated in a real galaxysample. Instead, we decided to set a fixed number density n(H)for each of our samples, and define Mmin in each model to matchit. This approach is in essence a very simple version of the haloabundance matching. As shown in Fig. 1, the halo mass functioncan be significantly different in f (R) models and in ΛCDM. Thismeans that, for each sample defined in this way, we may end upwith significantly different values of Mmin in each of our models.

We defined three halo populations for the present work,H1,H2,H3, with corresponding number densities n = 3 × 10−5,10−5 and 3 × 10−6 h3 Mpc−3, respectively. The upper limit for thenumber density, n(H1), was chosen based on the resolution lim-its of the simulations described above. The lower limit n(H3) waschosen to ensure that shot noise would not dominate our results.These three number densities were used to select the correspond-

MNRAS accepted, 1–17 (2017)

Halo clustering in f (R) cosmologies 7

10­8

10­7

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10­4

n(>M

h)[(h−

1M

pc)−

3]

z= 0 GRF6F5F4

0.50.00.51.01.5

(n/n

GR)−

1

z= 0. 25 GRF6F5F4

10­8

10­7

10­6

10­5

10­4

n(>M

h)[(h−

1M

pc)−

3]

z= 0. 66 GRF6F5F4

1013 1014 1015

Mh [h−1M¯]

0.50.00.51.01.5

(n/n

GR)−

1

z= 1. 00 GRF6F5F4

1013 1014 1015

Mh [h−1M¯]

Figure 1. Cumulative halo mass function of the different models considered in this work for the four different epochs z = 0, 0.25, 0.66, 1.0, as indicated. Thehorizontal dotted lines signal the number densities we use to define our three halo samples. In each case, the lower panel show the relative change with respectto the ΛCDM (GR) model.

ing samples at each of the redshift snapshots used. Table 2 lists thecorresponding values of Mmin used in each case. Following the dif-ferences in the mass function shown in Fig. 1, the values are nearlyidentical for the ΛCDM and F6 models, while it is larger for F5 andF4. As expected, in all cases, for a fixed n the corresponding Mmin

increases with decreasing redshift.The number densities of the selected halo samples can be

used to relate them to possible tracers to be used in the analysisof real surveys. The density of H1, for instance, is similar to thatof the brightest samples of luminous red galaxies (LRGs) typicallyused in the analysis of the Sloan Digital Sky Survey (SDSS, e.g.Martínez et al. 2009; Kazin et al. 2010). Lower densities as thoseof H2,H3 are typical of galaxy groups or clusters of varying rich-ness (Koester et al. 2007).

4 RESULTS

In this section we present and discuss our main results obtained forthe correlation function of dark matter and various halo samples atdifferent epochs, both in position and in redshift space. We studythe different components affecting the clustering of haloes separ-ately. In section 4.1 we study the clustering of the underlying matterdensity field, while in section 4.2 we analyse the correlation func-tion of haloes, derive the halo bias and assess its properties in thedifferent models.

4.1 Clustering properties of the matter density field

We first study the clustering of the smooth density field of the un-derlying matter component of our simulations. Although this stat-istic is not directly accessible via astronomical observations, it isnoteworthy to study the properties of ξm, since it can be related andinterpreted in a straightforward manner to the underlying theoret-ical model. Figure 2 presents the real-space correlation functionsof the dark matter distributions at the four redshifts considered. Ineach case, the black line and shaded area correspond to the meanvalue and 1σ scatter for the ΛCDM model. The corresponding scat-ter of the modified gravity runs is of the same order and scale-dependence as the fiducial ΛCDM case and hence it is not shownexplicitly in the plot for clarity. The different points and colour linescorrespond to the three f (R) models considered. The bottom panelsin each case show the relative difference of the three f (R) modelswith respect to the ΛCDM case.

We can already infer a number of interesting points from thedata shown in Fig. 2. Firstly, we observe that the amplitude of clus-tering grows on all scales monotonically with cosmic time. Thisis a well known result observed in all classes of cosmologies withhierarchical initial cold dark matter power spectra.

One important feature illustrated by Fig. 2 is the fact that thebaryon acoustic oscillations (BAO) peak scale is not affected bymodified gravity. It is apparent from the plot that all models showthis peak at a scale rpeak ' 110 h−1 Mpc. This corroborates our ex-

MNRAS accepted, 1–17 (2017)

8 Arnalte-Mur, Hellwing & Norberg

0

10

20

30

40

r2ξ m

(r)[(h−

1M

pc)

2]

z= 0 GRF6F5F4

0.30.20.10.00.10.20.3

(ξm/ξ

m,G

R)−

1

z= 0. 25 GRF6F5F4

0

10

20

30

40

r2ξ m

(r)[(h−

1M

pc)

2]

z= 0. 66 GRF6F5F4

0 20 40 60 80 100 120 140

r [h−1Mpc]

0.30.20.10.00.10.20.3

(ξm/ξ

m,G

R)−

1

z= 1. 00 GRF6F5F4

0 20 40 60 80 100 120 140

r [h−1Mpc]

Figure 2. Real-space correlation function of the matter density field, ξm(r), for the four models considered. For the GR case we plot both the mean value(black line) and the 1σ scatter (shaded area) over the 6 realizations. For the different f (R) models we only plot the corresponding mean values (colour linesand symbols, as indicated). The corresponding scatter is of the same order and scale-dependence as the GR one, hence we omit it for clarity. We plot thecorrelation function scaled by r2 to better visualize the function at large scales, where its amplitude is low. Each plot corresponds to a different epoch, asindicated. In each case, the lower panels show the relative differences with respect to the GR case.

pectation that the expansion history of the f (R) models is identicalto that of ΛCDM when the ratio c1/c2 is fixed according to eq. (8).However the position of the peak, and hence the cosmological in-formation that can be extracted from it, could in principle be af-fected by non-linear effects acting differently in GR and f (R) mod-els. To test this, we did a fit to our results using the simple modelcommonly used to analyse observations from galaxy redshift sur-veys (see, e.g., Anderson et al. 2014). This model accounts for thenon-linear damping of the BAO through the parameter ΣNL, andmeasures a possible change in the BAO scale with respect to thefiducial value through the parameter α. We find that we recover thecorrect value of α = 1 (and hence of the BAO scale) to within 2%without any significant difference between models. We do not findeither any significant difference for ΣNL, with values typically in therange ΣNL = 7 − 12 h−1 Mpc.

While the BAO peak scale is preserved, we can clearly noticein Fig. 2 that all four considered models experience growth of clus-tering that differ from each other, with differences varying in mag-nitude and scales at which they appear. At relatively early timesthe scalaron fifth force did not had enough time to significantly al-ter the growth of structures. This is clearly indicated by the resultsin the bottom-right panel, where at z = 1 all models show matterclustering consistent with each other. However as the cosmic evol-

ution progresses, we can observe a weak change of the correlationfunction amplitude in the f (R) models.

The F4 model at z = 0 manifests a large excess at r .35 h−1 Mpc when compared to ΛCDM and the two other f (R) mod-els. This is followed by a lower amplitude of ξm in the regime fromr & 50 h−1 Mpc up to the BAO peak. This behaviour reflects thefact that the f (R) models, and especially F4 (which is only veryweakly screened), are characterized by a scale-dependent growthrate f ≡ d ln D+/d ln a (Koyama et al. 2009). Such a strongly en-hanced matter clustering at small scales comes with a price of mat-ter that was more effectively evacuated from the interiors of largecosmic voids (Li et al. 2012b; Cai et al. 2015). The overall effectis very strong in F4, which is indicated by a significantly alteredslope of ξm at 20 . r/( h−1 Mpc) . 90. The F5 model shows a muchweaker discrepancy with respect to the GR case. There is a hint ofthe amplitude of ξm being lower than that of ΛCDM at scales ofr ∼ 60 h−1 Mpc, similar to the F4 case. At smaller scales, however,the F5 model results follow closely those of ΛCDM, as expectedfrom the stronger screening in this case. Similar behaviour (withweaker discrepancies) is found generally for the F4 and F5 modelsat z = 0.66 and z = 0.25. At z = 0.25 we see that the relative amp-litudes of the MOG models versus ΛCDM at intermediate scalesare slightly larger than expected from the global trends. Howeverthis is a small variation that could be due to a statistical fluctuation.

MNRAS accepted, 1–17 (2017)

Halo clustering in f (R) cosmologies 9

1.2

1.4

1.6

1.8

2.0

g(x)

z= 0 GRF6F5F4

0.100.050.000.050.10

(g/g

GR)−

1

z= 0. 25 GRF6F5F4

1.2

1.4

1.6

1.8

2.0

g(x)

z= 0. 66 GRF6F5F4

0 20 40 60 80 100 120 140

x [h−1Mpc]

0.100.050.000.050.10

(g/g

GR)−

1

z= 1. 00 GRF6F5F4

0 20 40 60 80 100 120 140

x [h−1Mpc]

Figure 3. Ratio of the redshift-space to the real-space matter correlation functions, g(x) =ξs

m(s=x)ξr

m(r=x) . As in previous figures, each plot corresponds to a differentepoch, and lower panels show the relative differences with respect to GR. The shaded area corresponds to the 1σ scatter for the GR case. The dotted lines inthe main panels illustrate the linear theory prediction gL(x) for each model. The horizontal black line shows the constant prediction for ΛCDM according toeq. (15), while the other lines show the scale-dependent predictions for F6 (blue), F5 (green) and F4 (orange) obtained using eqs. (16, 17).

The matter clustering of the F6 model is, at all redshifts, consistentwith the ΛCDM case.

To study the effect of redshift-space distortions in the matterdensity field, we plot in Fig. 3 the ratio of the redshift to real spacecorrelation functions g(x) ≡ ξs

m(s = x)/ξrm(r = x). The results here

can be compared to those of Jennings et al. (2012), who studiedthe effect of redshift-space distortions in f (R) cosmologies usingpower spectrum statistics for the same set of simulations as used inthis work.

In each case, we compare our results to the correspondinglinear-theory predictions. For the ΛCDM model, this correspondsto a constant ratio g, given by the Kaiser formula (Kaiser 1987;Hamilton 1992),

gGRL = 1 +

23

f +15

f 2 , (15)

where f , the linear growth rate,2 can be approximated in ΛCDMby f ≈ Ω0.55

M (z). For our cosmogony g ≈ 1.35 at z = 0, growingto g ≈ 1.69 at z = 1. For the case of the f (R) models, however,the growth rate depends on scale so the predicted ratio does alsodepend on scale. Koyama et al. (2009) computed the correspond-ing Fourier-space growth rates for each of our models as function

2 Not to be confused with the non-linear Lagrangian function f (R)

of wavenumber f (k). We use these to compute the configuration-space linear prediction gL(x) for each model as follows. First, fromthe real-space linear power spectrum Pr

L(k) we obtain the corres-ponding redshift-space power spectrum Ps

L(k) using the Kaiser for-mula,

PsL(k) =

[1 +

23

f (k) +15

f (k)2]

PrL(k) . (16)

We use the PrL(k) for each model and redshift obtained by Koyama

et al. (2009). We obtain the corresponding real- and redshift-spacelinear correlation functions using the standard Fourier transform

ξr,sL (x) = 4π

∫ +∞

0Pr,s

L (k)sin(kx)

kxk2dk(2π)3 , (17)

and compute the linear prediction for the g(x) ratio directly asgL(x) = ξs

L(s = x)/ξrL(r = x). In each panel of Fig. 3 we show

as dotted lines the linear-theory predictions calculated in this wayfor GR and our three MOG models.

Figure 3 illustrates that on scales x . 80 h−1 Mpc the ratiosg(x) for all models follow remarkably well the corresponding lin-ear predictions in each case. This may seem to be in contradic-tion with the results of Jennings et al. (2012) which showed clearlythe damping of the clustering due to virial motions at small scales(k & 0.05 h Mpc−1, see e.g. their fig. 4). However, note that inthis work we only consider scales x > 10 h−1 Mpc. In the case

MNRAS accepted, 1–17 (2017)

10 Arnalte-Mur, Hellwing & Norberg

of ΛCDM, we expect this damping effect to appear only at smal-ler scales in configuration space (see, e.g., Scoccimarro 2004). Ourresults indicate that this is the case also for the f (R) models.

The results for F5 and F6 in Fig. 3 agree to a good approx-imation with the measured g(x) for ΛCDM. This is due to the factthat the excess clustering predicted by the scale-dependent growthrates of f (R) appears typically at scales x . 20 h−1 Mpc for thesemodels – as shown by the linear theory predictions (dotted lines)–,while we study mostly larger scales. In fact, we can observe thatF5 deviates from GR at the smallest bin studied, x = 16 h−1 Mpc.F4 is clearly an outlier here, showing a clear enhancement in theratio g(x) with respect to GR at scales x . 50 h−1 Mpc. This wasto be expected: as we have already mentioned this model is nearlyunscreened, hence its growth rate is larger than the ΛCDM oneover a large range of scales. In the case of F5 and F6, on the otherhand, the screening mechanism makes the deviations in the growthfactor f (k) with respect to GR to appear only at smaller scales(k & 0.01 h Mpc−1), as shown by fig. 1 of Jennings et al. (2012).

The analysis of the redshift- to real-space matter correlationfunction ratios also reveals interesting behaviour around the BAOfeature. In real space, the BAO feature has the form of a relativelysharp peak in the correlation function centred at the BAO scale(see Fig. 2). In redshift space the peculiar velocities introduce ansmoothing of this BAO feature. This means that the amplitude nearthe centre of the peak is reduced, and this power is moved to thescales corresponding to the tails of the peak. When plotting the ra-tio g(x) as in Fig. 3 this results in the observed dip centred at theBAO scale (x ' 110 h−1 Mpc), and a peak at slightly smaller scales(x ' 90 h−1 Mpc). This behaviour is observed for the four models –GR and f (R)– considered. The apparent larger differences betweenall models that appear at the peak and dip scales are artificially en-hanced due to noise, since we take here a ratio of two very smallquantities. In this case, the noise is not expected to be Gaussian, sothe simple error estimation we used does not fully account for it.

4.2 Clustering of haloes

Now we turn to analyse the clustering properties of DM haloes.We computed the redshift-space correlation functions ξh(s) for ourthree halo populationsH1,H2 andH3 described in section 3.2. Weshow our results for the four redshifts considered in Fig. 4. Eachof the main panels show the correlation function for the three pop-ulations in the four gravity models considered. The lower panelsshow the relative difference for the three f (R) models with respectto ΛCDM, separately for each halo population. As explained in sec-tion 3, the ξh(s) we computed correspond to the statistic that can bemeasured from samples of luminous galaxies or galaxy groups andclusters in real observations. We could therefore compare directlyour theoretical results with observational data. Hence, any signific-ant difference we see in Fig. 4 can in principle serve as a way todiscriminate between ΛCDM and the f (R) models.

The four panels of Fig. 4 (corresponding to four different red-shifts) show the same main property of the clustering of our threehalo populations in the four models considered. The least abundanthalo samples (H3) show the highest amplitude of the correlationfunction, while inversely the highest number density sample (H1)exhibits the lowest ξh(s) amplitudes, with the intermediate samplelying in between. This is expected, as higher number density corres-ponds to lower (average) halo mass (see Table 2) and hence weakerclustering (lower bias parameter).

We focus now on the differences between models observed inthe different ξh(s). For the four considered epochs, the F6 model is

10­6 10­5 10­4

n [(h−1Mpc)−3]

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Hal

o bi

as

z= 0. 00

z= 0. 66

z= 1. 00

GRF6F5F4

Figure 6. Redshift-space halo bias estimated over the range s ∈

[24, 52] h−1 Mpc as function of the halo density used for selection of thepopulations. In addition to the samples used elsewhere in this work, we alsoshow for completeness the result for the sample with n = 10−6 h3 Mpc−3.The results for three different redshift snapshots, z = 0, 0.66 and 1.0, areshown as indicated by the labels. We omit here the results for z = 0.25 forclarity.

very close to the GR data. Hence the clustering of the three halopopulations is statistically indistinguishable in these two models.F4 and F5 show more pronounced differences in all cases. Thesedifferences can be appreciated more clearly (especially for F4) forpopulationsH1 andH2. This is partly due to the fact that for popu-lation H3 the number of tracers is low hence the statistical error isthe highest. In general, we observe that the halo correlation func-tions for these two MOG models are significantly lower than thecorresponding ΛCDM ones for scales s . 60−80 h−1 Mpc (or evenlarger scales in some cases for F4). It is interesting to note the dif-ference in behaviour of the F5 and F4 models. For F5 the departurefrom the GR signal consists typically of a global change in the amp-litude of ξh(s) for each halo population and redshift. This change inamplitude is visible for redshifts z > 0.25 but disappears at z = 0. Inthe case of F4 the deviation from GR seems to grow monotonicallywith time reaching the maximum at z = 0. Moreover, in addition tothe change in amplitude, we also observe for F4 a change in slopeat scales s . 60 h−1 Mpc with respect to the ΛCDM case. This dif-ference in slope is most clearly visible also at the lowest redshiftsz 6 0.25. These differences in the clustering of haloes across red-shifts reflect most likely a combination of many different effects:variations in the overall matter clustering (Fig. 2), discrepancies inthe magnitude of the redshift space distortions (Fig. 3) and finally adeviation in halo bias (see eq. 12). To infer deeper into this, belowwe study the halo bias characteristics of our halo populations andthe differences between our models.

Following eq. (12), we measure the redshift-space halo biasfrom the simulations using the estimator

bh(s) =

√ξh(s)ξm(s)

. (18)

We focus here only on the redshift-space bias as this is the theor-etical quantity relevant for comparison with typical observationalmeasurements of ξ in galaxy surveys. However our results for thereal-space bias are very similar to the ones presented here, withonly a global change in the amplitude.

Figure 5 presents the bias as a function of scale s for our three

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Halo clustering in f (R) cosmologies 11

100

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z= 1. 00 GRF6F5F4

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H1

H2

H3

Figure 4. Redshift-space correlation functions of the different halo samples considered, ξh(s), for our four models. As in Fig. 2, the functions amplitudes wererescaled by s2. As in previous figures, each plot corresponds to a different redshift and the shaded area corresponds to the 1σ scatter for the GR case. In themain panels the different groups of lines correspond, from bottom to top, to the halo populationsH1,H2 andH3. The three lower panels in each plot show therelative differences with respect to GR for the indicated halo population.

halo populations. We only plot b(s) for s 6 110 h−1 Mpc to avoidscales where either ξm(s) or ξh(s) become negative. As a comple-mentary plot we also show in Fig. 6 the bias, averaged over a rangeof scales s ∈ [24, 52] h−1 Mpc, as a function of the number densityof the halo population. We chose these scales as in that range thebias is reliably measured and approximately constant in the case ofΛCDM, as shown in Fig. 5. At smaller scales the non-linear evol-ution of the density and velocity fields becomes important and asimple linear bias description breaks down. The rapidly increasingb(s) value observed at s . 20 h−1 Mpc is a hint of this non-linearbehaviour. The weak scale dependence observed at large scales,where s > 80 h−1 Mpc, may be also due to non-linear effects nearthe BAO peak. However, at these scales the scatter is large due tocosmic variance and low amplitudes of both matter and halo ξ(s),so these effects are not statistically significant here.

The f (R) halo samples are characterized always by a bias thatis either smaller or equal to the fiducial GR case. If we recall that inf (R) haloes tend to be, on average, more massive than in GR, this

result may seem surprising. If we look at a population of haloesat fixed virial mass, in f (R) there will be many haloes that origin-ate from smaller density peaks than their equivalent z = 0 masscousins in GR. Since the initial conditions are the same within theensemble, haloes that originate from smaller peaks (lower Jeansmass), which are characterized by lower bias, are in f (R) shifted to-wards higher masses and then compared with the fiducial GR casethat originate from rarer peaks (hence higher bias). This is con-sistent with the picture seen in Fig. 6, where we observe that differ-ences in bias are higher for higher redshift, just as the differences inmass functions in Fig. 1. Another feature of the f (R) halo bias seenin Fig. 5 is its stronger scale dependence than in the ΛCDM case.This can be especially seen for F4, where at 20 6 s/( h−1 Mpc) 6 60the bias is increasing with scale. Similar but much weaker beha-viour can be also observed for F5 at z = 0.66.

Once we have studied the differences in halo bias, we can bet-ter interpret the differences in the halo correlation function ξh(s)observed between the f (R) models F4 and F5 and the ΛCDM case

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12 Arnalte-Mur, Hellwing & Norberg

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Figure 5. Redshift-space halo bias as function of scale s for different halo populations. As in previous figures, each plot corresponds to a different redshift andthe shaded area corresponds to the 1σ scatter for the GR case. In the main panels the different groups of lines correspond, from bottom to top, to the bias ofhalo populationsH1,H2 andH3. The three lower panels in each plot show the relative differences with respect to GR for the indicated halo population.

(Fig. 4). For the case of the F4 model, it is clear that the steeperslope and excess clustering at small scales observed for both ξm(r)(Fig. 2) and g(x) (Fig. 3) is compensated by a significantly smal-ler bias (and positive scale dependence). This results in the ξh(s)having in all cases a smaller amplitude than the ΛCDM case, butwith only a mild change of slope (except at z = 0). A similar ex-planation can be given for the halo correlation functions in the F5model, although the differences with respect to ΛCDM in this caseare smaller.

5 OBSERVATIONAL TESTS USING CLUSTERINGSTATISTICS

The results shown in section 4.2 indicate that for both F4 and F5halo samples of the same number density n can be characterizedtypically by significantly different values of the correlation func-tion ξh(s) than our fiducial GR model. In principle, as our halosamples can be directly related to samples of luminous galaxies

or groups in real surveys (with the caveats discussed in section 3),these ξh(s) are observable quantities. Hence, they could be used todiscriminate between GR and these MOG models. However, thedifferences in ξh(s) seen in Fig. 4 could be degenerate with changesin the ΛCDM clustering due to variations of the cosmological para-meters, and in particular σ8. Therefore, one would need to combinethe ξh(s) measurements with other model-independent determina-tions of these parameters. An alternative would be to measure dir-ectly the halo bias and use the differences between models seen inFigs. 5 and 6. Bias can not be directly obtained from two-point stat-istics, but there exist estimates based on weak lensing observations(McKay et al. 2001; Covone et al. 2014; van Uitert et al. 2016)and on higher-order statistics of the galaxy distribution (e.g. Verdeet al. 2002; Gaztañaga et al. 2005; McBride et al. 2011; Arnalte-Mur et al. 2016). However, these methods infer bias from obser-vations in a model-dependent way, hence all the systematic effectswere checked only against the assumed ΛCDM cosmology.

In this section we explore a way in which we can nevertheless

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Halo clustering in f (R) cosmologies 13

use the two-point clustering of haloes to test observationally thestudied f (R) models. We try to define a statistic based on the clus-tering of haloes that: (i) can differentiate between ΛCDM and dif-ferent f (R) models, following our results in section 4.2, and (ii) canbe measured from observations in a way which is as model inde-pendent as possible (i.e. does not depend on the clustering proper-ties of the matter density field). As the differences between modelsobserved above vary with both scale and halo population, the bestway to define such a statistic is to combine the correlation functionsξh(s) for different populations and at different scales.

Hence, we define the relative clustering ratio R for a halo pop-ulationH as function of scale s as

R(s,H|Href , sref) =s2ξh(s|H)

s2refξh(sref |Href)

, (19)

where Href is a reference halo population and sref is a referencescale (kept fixed). Here we use the term s2/s2

ref to rescale the cor-relation functions in order to have comparable values as a functionof sref . As we show below, this new statistic can be predicted theor-etically for each model using the results of section 4.2. It can alsobe directly measured from observations – with the caveats men-tioned in section 3 to identify a halo population with a class ofobserved objects. The way to compute R in a given survey is tofirst identify the relevant populations equivalent toH andHref andobtain the corresponding catalogues of objects. One then computesthe redshift-space correlation function for each of these cataloguesusing an standard estimator (e.g. Landy & Szalay 1993). The clus-tering ratio R is finally computed using directly eq. (19) above. Inthis way, R will be independent of the amplitude of the matter cor-relation function, σ8. Furthermore, as both populationsH andHref

are extracted from the same volume (same survey), the effects ofsampling variance of the R ratio will be additionally suppressed.

Here, we choose for the reference population Href = H1, thesample with the highest spatial abundance, n = 3 × 10−5 h3 Mpc−3.The scale-dependent differences between models in ξh(s) can ap-pear in different ways in R depending on the value of sref used.Therefore sref can be chosen, in principle, to maximize the differ-ences between models. Here, we show our results for two referencescales: sref = 16 and 64 h−1 Mpc. These two values were chosen tospan the range of scales where the discrepancies are more clearlyobserved in Fig. 4, while avoiding larger scales where errors cangrow significantly.

In Figs. 7 and 8 we plot the clustering ratio R for our tworeference scales, sref = 16 h−1 Mpc and sref = 64 h−1 Mpc respect-ively. The clustering ratio R for theH1 population is a special case,as this is the population we use as reference for our calculations.In this case R(s) is just the halo correlation function ξh(s) normal-ized to its amplitude at s = sref . As the difference between the halocorrelation functions of F5 and GR was just a constant shift in theamplitude, this difference completely disappears in the case of R.For the F4 model, however, this difference with respect to GR hasa dependence on scale, and therefore we also see a significant de-viation in R at z = 0, 0.25 for both values of sref .

When we consider different samples (H2 and H3) the situ-ation changes, as here the R depends also on the relative biasbetween different populations. For sref = 16 h−1 Mpc (Fig. 7), F5presents some departures from GR, exhibiting lower values of Rfor s . 40 h−1 Mpc at z 6 0.66. These departures are small (onlya few per cent), but significant for H2. As these scales (and sref)correspond to the mildly non-linear regime, this could be due tothe non-linear effects scaling differently with halo mass in F5 andGR. This discrepancy could be used, in principle, to discriminate

between the F5 model and ΛCDM. However, given the small sizeof the effect, this would be difficult in practice due, e.g., to possiblesystematic errors.

Moving to the larger reference scale sref = 64 h−1 Mpc (Fig. 8)the results for F5 are completely consistent with GR. This indic-ates that the F5 signature at linear scales is reduced to a globalchange in the amplitude of clustering. On the other side, we obtainhere deviations from GR that are large and statistically signific-ant for the least screened f (R) model, F4. These deviations growwith decreasing redshift, attaining relative changes of ∼ 20% at thesmallest scales for all halo populations. For z 6 0.25 the statist-ical significance of these deviations is ∼ 2− 5σ. This indicates thatR(s,H|H1, sref = 64 h−1 Mpc) can be used to render constraints forstrongly deviating models like F4. As expected from all our previ-ous results, for all the considered snapshots and reference scales,the R of F6 are statistically consistent with GR.

6 DISCUSSION AND CONCLUSIONS

We have analysed the real- and redshift-space two-point clusteringstatistics of DM and haloes in a series of simulations employing thestructure formation in the ΛCDM and f (R) cosmological models.We have also introduced a new statistic - the halo relative cluster-ing ratios R(s,H|Href , sref). We have fixed our analysis on threehalo populations constructed by implementing fixed number dens-ity cuts at n = 3 × 10−5, 10−5 and 3 × 10−6 h3 Mpc−3 (denoted,respectively, H1,H2 and H3). Hence our halo populations mimicin a general sense spatial selection effects similar to those found involume-limited samples from redshift galaxy surveys. The numberdensities we use are typical of samples of very luminous galaxies,or of groups and clusters of galaxies. We can summarize our mostimportant findings in the following points:

• In all models the clustering amplitude of DM grows mono-tonically with time. At high redshifts the matter clustering is indis-tinguishable among models. At later times (z . 0.66) our strongestmodel - F4 - shows significant deviations of the amplitude and slopeof ξm at small and intermediate scales, while ξm of both F5 and F6remain mostly consistent with ΛCDM. In all models the BAO peakscale is the same and is not affected in any significant way by thefifth force;• The ratio, g(x), of redshift- to position-space matter correl-

ation functions of F5 and F6 is compatible at large-scales (x >25 h−1 Mpc) with the ΛCDM results. F4 is a strong outlier here,showing significant deviations up to x ∼ 50 h−1 Mpc. All four mod-els show good agreement with the respective linear theory predic-tions in the range 15 . x/( h−1 Mpc) . 80.• The differences of the redshift-space two-point correlations of

haloes are bigger than in the case of the DM density field. In gen-eral, the halo correlation functions of the F4 and F5 models arelower than those of GR. This is more clearly observed for the H1

and H2 samples, because of the larger errors in H3 (due to sparsesampling). The strong F4 model is an outlier at all epochs, with thestrongest signal at z = 0. However, for the F5 model the ξh reachesits maximal departure from GR at intermediate and high redshifts,z > 0.25.• Halo bias in all f (R) models and for all halo populations is al-

ways lower than in GR or consistent with the fiducial model. AgainF4 is an outlier here at all scales and epochs. The F6 model halobias is fully consistent with the GR predictions, while for F5 themost significant differences appear again at intermediate and highredshifts.

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14 Arnalte-Mur, Hellwing & Norberg

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Figure 7. Relative clustering ratio R(s,H|Href , sref ) for different halo populations, for the case in which the reference sample isHref = H1, and the referencescale is set to sref = 16 h−1 Mpc. As in previous figures, each plot corresponds to a different redshift and the shaded area corresponds to the 1σ scatter for theGR case. In the main panels the different groups of lines correspond, from bottom to top, to the clustering ratio obtained for halo populationsH1,H2 andH3.The three lower panels in each plot show the relative differences with respect to GR for the indicated halo population.

• Finally we considered the relative clustering ratios R to con-struct a largely model-independent observational clustering probe.The F4 model halo clustering ratios depart significantly from theGR model for all our samples, specially when using as referencescale sref = 64 h−1 Mpc and at z 6 0.25. Again F6 is characterizedby too small differences from GR to be statistically distinguishablein any way. However the R of the mild F5 model at redshifts ofz 6 0.66 and for sref = 16 h−1 Mpc is showing a small but signific-ant signal at scales s . 40 h−1 Mpc.

Our results indicate that only in the case of the unrealisticallystrong and not screened F4 model one can expect a clear, strongand significant signal visible in both the matter and halo cluster-ing. This signal for F4 is also clear in the clustering ratios R. Thismeans that this model could be tested using only the two-point clus-tering of haloes in a model-independent way. On the other end ofthe spectrum the highly screened F6 model is always very close toGR for all our statistics and samples and at all epochs. Hence, these

models are indistinguishable from each other, at least when one isconcerned with the two point clustering statistics. For the physic-ally interesting F5 model we have found only small differences withrespect to GR in the clustering ratios R. It shows, however, a signi-ficant signal in the raw halo correlation functions ξh(s), that can besummarized as changes in a constant linear bias as function of halopopulation and redshift (Fig. 6). The predicted signal is strongestfor redshifts z > 0.25. Two-point clustering observations can not beused to measure the bias on their own. However, our results sug-gest that they could be used in combination with other probes (e.g.an independent measurement of σ8) to put constraints on the F5model.

Our results yield the hope that growing observational data willbe able to constrain this class of f (R) models using galaxy clus-tering. Two near-term projects that may have the potential to per-form these tests are the DESI (Levi et al. 2013; DESI Collabora-tion et al. 2016) and J-PAS (Benitez et al. 2014) surveys, whichwill cover a large fraction of the sky (14000 deg2 and 8500 deg2,

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Halo clustering in f (R) cosmologies 15

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Figure 8. Same as Fig. 7, for the case in which the reference scale is set to sref = 64 h−1 Mpc.

respectively). Both projects will target different classes of galax-ies up to redshifts z . 1, therefore covering the range of redshiftsstudied in this work. Given the expected number density, it will bepossible to select samples of galaxies (e.g. LRGs) that can be re-lated to the halo populations we studied. It will also be possible touse for these tests catalogues of galaxy groups and clusters fromthese surveys. Ascaso et al. (2016) showed that it will be possibleto detect reliably in J-PAS clusters corresponding to halo masses ofM & 3.6 × 1013 h−1 M up to z ' 0.7 (assuming a ΛCDM cosmo-logy). This selection would match ourH1 sample at the lowest red-shifts. Slightly further in the future, another survey suitable for thistype of analysis will be Euclid (Laureijs et al. 2011). Euclid willobserve galaxies over a significantly larger volume, thus reducingthe statistical error of the clustering measurements. However, itsspectroscopic survey will be limited to higher redshifts than thosestudied in this work (z & 0.9), where we expect the differences inclustering between the GR and f (R) to be smaller.

We therefore expect that our method will be able to constrainthe particular Hu & Sawicki (2007) model considered here downto | fR0| ' 10−5 using data from these near-future surveys. This

is competitive with possible constraints using other known meth-ods. DESI Collaboration et al. (2016), for example, forecast thatin the ideal case DESI will be able to measure the growth rate fat scales k 6 0.1 h Mpc−1 to a precision of ' 2 − 4% for redshiftsz ∈ [0.6, 1.0] (see their tables 2.3 and 2.4). For comparison, Jen-nings et al. (2012) show that the maximum expected change in fwith respect to GR at these scales and redshifts is ' 5% (' 1%) for| fR0| ∼ 10−5 (| fR0| ∼ 10−6). Therefore, this type of measurementscould yield constraints of the same order as those achievable usingthe clustering ratios R. Alternatively, Cataneo et al. (2015) predictthat it will be possible to obtain even better constraints when futuresurveys allow for the detailed measurement of the cluster and groupmass function to higher redshifts (z ∼ 2).

Our analysis of the g(x) ≡ ξsm(x)/ξr

m(x) ratios showed, how-ever, that one needs to take caution when trying to extract thegrowth rate f from just the halo/galaxy/matter clustering signal.Although the scale-dependent growth rates predict an enhancementin the redshift-space clustering in f (R) models, this is only seen atrelatively small scales, x < 20 h−1 Mpc (x < 50 h−1 Mpc in F4). Atthese small scales, we will find deviations from linear theory due

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16 Arnalte-Mur, Hellwing & Norberg

to the effect of virial motions, and one should take into account thepredicted enhanced peculiar velocities in MOG models (Zu et al.2014; Hellwing et al. 2014; Sabiu et al. 2016). That means that,in order to use the growth rate inferred from galaxy redshift cata-logues to constrain this class of models, it is necessary to modelthese effects in detail, and to test the analysis method with realisticMOG mocks (Barreira et al. 2016). This problem is largely alle-viated in the case where we only consider redshift-space relatedquantities, such as the halo correlation function ξh(s) or speciallythe clustering ratios R, and so avoid the necessity of modelling pre-cisely the connection between position and redshift space objects.In sum we advertise here to use R(s,H|Href , sref) to study and con-strain f (R) models using the redshift space clustering of galaxiesas measured by modern galaxy surveys.

ACKNOWLEDGEMENTS

We thank the referee, Dr Nelson Lima, for his comments thathelped improve the clarity of the paper. The authors are very grate-ful to Baojiu Li for many inspiring comments and for providingcosmological simulations used in this work. We have also benefitedfrom comments and discussions we had with Shaun Cole, CarlosFrenk, Kazuya Koyama, Will Percival and Violeta González-Pérez.WAH is grateful to the co-authors for the endless patience theyshowed during writing of this draft. PAM was supported by theEuropean Research Council Starting Grant (DEGAS-259586) andby the Generalitat Valenciana project PrometeoII 2014/060, and ac-knowledges additional support from the Spanish Ministry for Eco-nomy and Competitiveness through grants AYA2013-48623-C2-2and AYA2016-81065-C2-2-P. WAH acknowledges support fromthe European Research Council grant through 646702 (CosTes-Grav) and the Polish National Science Center under contract#UMO-2012/07/D/ST9/02785. PN acknowledges the support ofthe Royal Society through the award of a University ResearchFellowship, the European Research Council, through receipt of aStarting Grant (DEGAS-259586) and the Science and TechnologyFacilities Council (ST/L00075X/1). This work used the DiRACData Centric system at Durham University, operated by the Insti-tute for Computational Cosmology on behalf of the STFC DiRACHPC Facility (www.dirac.ac.uk). This equipment was funded byBIS National E-infrastructure capital grant ST/K00042X/1, STFCcapital grant ST/H008519/1, and STFC DiRAC Operations grantST/K003267/1 and Durham University. DiRAC is part of the Na-tional E-Infrastructure. This research was carried out with the sup-port of the HPC Infrastructure for Grand Challenges of Science andEngineering Project, co-financed by the European Regional De-velopment Fund under the Innovative Economy Operational Pro-gramme. This research made use of matplotlib, a Python libraryfor publication quality graphics (Hunter 2007).

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This paper has been typeset from a TEX/LATEX file prepared by the author.

MNRAS accepted, 1–17 (2017)